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5.3 Ideal and real filters
Sometimes, for reasons of standardization and simplification, it is convenient to treat the active filters to approximate way, upgrading them to
idealized and simplified theoretical models which are called ideal filters. In still other cases, the use of these standards are insufficient or lead to remarkable errors then the filter should be treated based on accurate real behavior, ie as a real filter.
5.3.1 Ideal Filters
Ideal is a filter that meets the following 4 key terms:
• A gain (amplification) units, and neither create nor strengthen
degradation of the input signal to the full extent of the zone or passing.
• Creates a complete degradation (100%) of the input signal across the bands, or ex.
• The transition of response from one zone to another is quite abrupt.
• Does not create any distortion to signals passing through the transit zones.
The above lead to the Sch.5.7 response curves for the four main categories of ideal filter, ie FTT (a), PHY (b), FZD (c) and FZA (d). As observed, all these ideal curves are orthogonal and the voltage gain in the passband is unique.
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Figure 5.7. Response curves of ideal filters: FTT (a), PHY (b), FZA (c), FZA
(D)
In the ideal FTT, the passband extends from zero frequency to the cutoff frequency f1. In an ideal PHY, the passband extends from the cutoff frequency f2 to the infinite frequency. In the ideal FZD, the transit zone covers the area f1 <f <f2. Finally, the ideal FZA, the passband extends from zero frequency up to f 1 and f2 by up to infinite frequency.
5.3.2 Real Filters
The actual filters have behavior similar to that provided by the ideals filters only very approximate. Thus, for example the sharp transition from passband to stopband and vice versa, is considered the ideal filter is not feasible in real filters. Also feasible is the stability of the voltage gain value throughout its extent or dielefsis.Telos zones, the existence of zero aid throughout its extent or cut areas are impossible in real filters.
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Usually, in contrast with the ideal response curves Sch.5.7,
practice the standard response curve of a real filter
Specifically an FTT, the pattern depicts the Sch.5.8 (with voltage gain 1).
Figure 5.8. Practical response curve of a real FTT
From this figure we can note the following singular
Response characteristics of a real filter.
• The endpoint of the pass band appears local aid reduction in price gain below 0 dB. This reduction is small but not negligible and has a maximum value denoted by Amax.
• In the relegation zone cutoff of aid is large but not infinite (as in the ideal filter). Usually, in this zone and near the cutoff frequency observed flare response curve. As a result, the degradation of the gain shows a minimum value denoted by Amin.
• Between the pass band and the stopband shows a transition zone in which the voltage gain decreases gradually rather than abruptly. This transition zone extends between the cutoff frequency f1 and another frequency fs.
How steep is the decline of the response curve in the transition zone depends on the frequency ratio fs / f1. If the transition is very sharp
we have fs = f1 so fs/f1 = 1.
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The more "gentle" is the transition, ie the smaller is the slope of the response curve in the transition zone, greater than 1 will be the reason.
We should note too that the above curve, a logarithmic scale of frequency depends on the so-called order or grade of the filter. For the exact meaning of the order of a filter will return below.
5.4 Families of filters
In mathematical analysis of the filters have been developed various mathematical models (standards) and circuits that are trying to achieve a best possible simulation of the behavior of ideal filters. These models are classified into the following four families: Filters Butterworth, filters Chebyshev, Bessel filters and Elliptic filters.
The Sch.5.9 depicts the typical response curves of these filters.
Figure 5.9. Family filter response curves
Butterworth filters are standard filters which prosomeionoun well the decline of the response curve in the transition zone, but do not show the ripple of the gain observed in the passband and stopband of the other filters. In these filters, the response in the passband appears consistently flat throughout its extent (megistoepipedi).
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In these filters, a bottom edge of the transition zone, ie a frequency f1, is the frequency at which (the actual filter) the height of the response curve, provided by the measure of the transfer function falls to the level of 0 dB,
by Amax, Sch.5.8. The other end of the transition zone is the frequency fs for the level of the response curve during Amin falls from the level of 0 dB (the beginning of the stopband).
The filters Chebyshev, unlike the previous ones, enabling the existence and calculation of ripple in the passband. They have, however, afxomoiosi the gain in the transition zone and steeper than the filters Butterworth.
Bessel filters do not show the ripple-response curve and the transition of migration gain is worse than in other families of filters. Their advantage is that they have linear phase and are used in systems that process pulses because of the condition, does not alter the shape of the pulse.
Finally, the elliptic filters have ripple in the response curve in both the passband and in the stopband, while the migration gain in the transition zone is steeper than in all other families of filters, Sch.5.8.
5.5 Degree or a filter order
In the filter transfer function H (s) or H (jf) is expressed by a performance in which the denominator is a polynomial in s and j f. O (algebraic) degree of this polynomial is called the grade or order, n, of the filter. In this book we will study first filters (n = 1) and second (n = 2) order.
In practice, sizes Amax <, Amin, f <fi and fs confounding the degree of the filter, which we saw in section 5.3. In paragmatikotita, n directly determines the slope (k) of the transition region (transition) of the filter response curve, Sch.5.10. In logarithmic frequency scale, the slope is related to the extent of the filter based on an approximation formula
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k = + (dB / decade) = + 6 n (dB / octave)
k = -20 db / Dec. (N = 1) n = -40 db / Dec. (N = 2)
Figure 5.10. Grade and slope filter
Noted that ten is the interval between two frequencies from which the largest is 10-fold the smallest, while octave is the interval in which the higher frequency is 2-fold the less. Besides, the (-) indicates the Ex.5.5.1 negative slope, thus descending curve (if FTT) and (+) positive slope, so rising curve (if PHY).
Therefore, when Ex.5.5.1 in a first order FTT slope is k = 20 1 = - 20dB/dekada the - 6 dB / octave (line "1" shape), while a second order PHY, the slope is k = + 20 2 = + 40 dB / decade = + 12 dB / octave.
The Sch.5.11 shows the circuit of an active FTT first aid order with Mr. O operational amplifier (TE) is syndesmologimenos as voltage controlled amplifier (VCVS) but called Sallen-Key filter is first order in honor of the two researchers developed.
5.6 FTT first order
It turns out that the filter transfer function is given by
the relation:
K = 1
R1
is to increase (voltage gain) at dc, ie f = 0 and 1
cutoff frequency of the filter. The frequency is one for
which is why f/f1 In Eq. (5.6.1) becomes unit, so the measure of
is the transfer function | H BW) I = K / / 2 <0.707 K, the 20 log K - 3dB,
ie 3 dB below the maximum gain (in dB).
O calculation of this filter, made with the help of
Eq. (5.6.2) and (5.6.3), metaschimatizomenes to get below
form:
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R2
Figure 5.11. Sallen-Key FTT first order
(5.6.5)
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To calculate the filter, given the strengthening K and cutoff frequency f1, we (even arbitrarily) an appropriate value for the capacity C and an appropriate value of R1 and then calculate the resistances R and R2.
Example 5-1
We compute a first order FTT, Sallen-Key type of support 5 and cutoff frequency 3000 Hz.
Solution
We C = 10 nF and R1 = 10 KO. From Eq. (5.6.4) and (5.6.5), we find:
R = 1 = 5.3 kQ <R = 5.6 kO (E12 series)
2 pi x 3000 x 10-8
and R2 = 4 x 10 kO = 40 kO or R2 = 39 kO
(The final values of the resistors are selected from E12 series). Sch.5.12 The curve shows the response of the filter calculated.
Figure 5.12. FTT with f = 3000 Hz and K = 5-response curve
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5.7 PHY first order
If the position permutations of the capacitor C and resistance R in the circuit Sch.5.11 shows PHY Sallen-Key, Sch.5.13. It turns
that the transfer function of this filter is given by:
(5.7.1)
where, f 2 = 1/2p RC
(5.7.2)
is the cutoff frequency.
The voltage gain K is still given by Eq. (5.6.2).
Figure 5.13. FTT first order
O calculation of the filter is just as relevant to the FSF.
Example 5-2
PHY to calculate a first order, Sallen-Key type with K = 5 and
f = 300 Hz.
Solution
We, C = 100 nF and R = 10 kQ. Therefore we have:
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and R2 = 4 x 10 kO = 40 kO L R2 = 39 kO (Series E12)
The Sch.5.14 the response curve of the filter calculated (as shown by the simulation program mCap III).
Figure 5.14. PHY with K = 5 and f2 = 300 Hz: Response Curve
